Large Monochromatic Components in Two-colored Grids

Abstract

Let $D_n^d$ denote the d-dimensional grid with diagonals, that is, the graph with vertex set ${\{1,2,\dots ,n\}}^d$ and with edges connecting every two vertices that differ by at most 1 in every coordinate. We prove that for an arbitrary 2-coloring of the vertices of $D_n^d$ there exists a monochromatic connected subgraph with at least $n^{d-1}-d^2{n}^{d-2}$ vertices. We also consider a d-dimensional triangulated grid; this is the graph of a triangulation of the solid cube ${[1,n]}^d$ that refines the subdivision of ${[1,n]}^d$ into the grid of unit cubes. Here every 2-coloring has a monochromatic connected subgraph with $\mathrm{\Omega }(n^{d-1}/\sqrt{d})$ vertices.